**Applied mathematics** is the application of __mathematical methods__ by different fields such as __science__, __engineering__, __business__, __computer science__, and __industry__. Thus, applied mathematics is a combination of __mathematical science__ and specialized knowledge. The term "applied mathematics" also describes the __professional specialty__ in which mathematicians work on practical problems by formulating and studying mathematical models. In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in __pure mathematics__ where abstract concepts are studied for their own sake. The activity of applied mathematics is thus intimately connected with research in pure mathematics.

__ Computational Mathematics__ involves

**Game Theory**

**Game theory**is "the study of__mathematical models__of conflict and cooperation between intelligent rational decision-makers". Game theory is mainly used in__economics__,__political science__, and__psychology__, as well as in__logic__and__computer science__.__[1]__Originally, it addressed__zero-sum games__, in which one person's gains result in losses for the other participants. Today, game theory applies to a wide range of behavioral relations, and is now an__umbrella term__for the science of logical decision making in humans, animals, and computers.*Cooperative / Non-cooperative**Symmetric / Asymmetric**Zero-sum / Non-zero-sum**Simultaneous / Sequential**Perfect information and imperfect information**Combinatorial games**Infinitely long games**Discrete and continuous games**Differential games**Many-player and population games**Stochastic outcomes (and relation to other fields)**Metagames**Pooling games**Mean field game theory*

**Fluid Dynamics**

n**I**__physics__and__engineering__,**fluid dynamics**is a subdiscipline of__fluid mechanics__that describes the flow of__fluids__-__liquids__and__gases__. It has several subdisciplines, including__aerodynamics__(the study of air and other gases in motion) and**hydrodynamics**(the study of liquids in motion). Fluid dynamics has a wide range of applications, including calculating__forces__and__moments__on__aircraft__, determining the__mass flow rate__of__petroleum__through__pipelines__,__predicting weather patterns__, understanding__nebulae__in__interstellar space__and__modelling fission weapon detonation__.**Numerical Analysis**

**Numerical analysis**is the study of__algorithms__that use numerical__approximation__(as opposed to general__symbolic manipulations__) for the problems of__mathematical analysis__(as distinguished from__discrete mathematics__).**Optimization**

In__mathematics__,__computer science__and__operations research__,**mathematical optimization**or**mathematical programming**, alternatively spelled*optimisation*, is the selection of a best element (with regard to some criterion) from some set of available alternatives.*Convex programming**Integer programming**Quadratic programming**Fractional programming**Nonlinear programming**Stochastic programming**Robust programming**Combinatorial optimization**Stochastic optimization**Infinite-dimensional optimization**Heuristics**and**metaheuristics**Constraint satisfaction**Disjunctive programming**Space mapping**Calculus of variations**Optimal control**Dynamic programming**Mathematical programming with equilibrium constraints**Multi Objective Optimization**Multi Modal Optimization*

**Probability Theory**

**Probability theory**is the branch of__mathematics__concerned with__probability__. Although there are several different__probability interpretations__, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of__axioms__. Typically these axioms formalise probability in terms of a__probability space__, which assigns a__measure__taking values between 0 and 1, termed the__probability measure__, to a set of outcomes called the__sample space__. Any specified subset of these outcomes is called an__event__.**Statistics**

**Statistics**is a branch of__mathematics__dealing with the collection, analysis, interpretation, presentation, and organization of__data__. In applying statistics to, for example, a scientific, industrial, or social problem, it is conventional to begin with a__statistical population__or a__statistical model__process to be studied. Populations can be diverse topics such as "all people living in a country" or "every atom composing a crystal". Statistics deals with all aspects of data including the planning of data collection in terms of the design of__surveys__and__experiments__**Cryptography**

**Cryptography**or**cryptology**(from__Ancient Greek__:__κρυπτός__,__translit.__*kryptós*"hidden, secret"; and__γράφειν__*graphein*, "to write", or__-λογία__, "study", respectively) is the practice and study of techniques for*-logia*__secure communication__in the presence of third parties called__adversaries__. More generally, cryptography is about constructing and analyzing__protocols__that prevent third parties or the public from reading private messages; various aspects in__information security__such as data__confidentiality__,__data integrity__,__authentication__, and__non-repudiation__are central to modern cryptography. Modern cryptography exists at the intersection of the disciplines of__mathematics__,__computer science__,__electrical engineering__,__communication science__, and__physics__.**Mathematical Finance**

**Mathematical finance**, also known as**quantitative finance**, is a field of__applied mathematics__, concerned with mathematical modeling of__financial markets__. Generally, mathematical finance will derive and extend the__mathematical__or__numerical__models without necessarily establishing a link to financial theory, taking observed market prices as input. Mathematical consistency is required, not compatibility with economic theory. Thus, for example, while a__financial economist__might study the structural reasons why a company may have a certain__share price__, a financial mathematician may take the share price as a given, and attempt to use__stochastic calculus__to obtain the corresponding value of__derivatives__of the__stock__(*see:**Valuation of options**;*). The*Financial modeling*__fundamental theorem of arbitrage-free pricing__is one of the key theorems in mathematical finance, while the__Black–Scholes__equation and formula are amongst the key results.**Mathematical Physics**

**Mathematical physics**refers to the development of mathematical methods for application to problems in__physics__. Thedefines the field as "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories". It is a branch of*Journal of Mathematical Physics*__applied mathematics__, but deals with physical problems.**Mathematical Chemistry**

**Mathematical chemistry**is the area of research engaged in novel applications of__mathematics__to__chemistry__; it concerns itself principally with the__mathematical modeling__of chemical phenomena. Mathematical chemistry has also sometimes been called**computer chemistry**, but should not be confused with__computational chemistry__. Major areas of research in mathematical chemistry include__chemical graph theory__, which deals with__topology__such as the mathematical study of__isomerism__and the development of__topological descriptors__or indices which find application in__quantitative structure-property relationships__; and chemical aspects of__group theory__, which finds applications in__stereochemistry__and__quantum chemistry__.**Mathematical Biology**

**Mathematical and theoretical biology**is a branch of__biology__which employs theoretical analysis, mathematical models and abstractions of the__living organisms__to investigate the principles that govern the structure, development and behavior of the systems, as opposed to__experimental biology__which deals with the conduction of experiments to prove and validate the scientific theories. The field is sometimes called**mathematical biology**or**biomathematics**to stress the mathematical side, or**theoretical biology**to stress the biological side. Theoretical biology focuses more on the development of theoretical principles for biology while mathematical biology focuses on the use of mathematical tools to study biological systems, even though the two terms are sometimes interchanged**Mathematical Economics**

**Mathematical economics**is the application of mathematical methods to represent theories and analyze problems in__economics__. By convention, the__applied methods__refer to those beyond simple geometry, such as differential and integral__calculus__,__difference__and__differential equations__,__matrix algebra__,__mathematical programming__, and other__computational methods__. An advantage claimed for the approach is its allowing formulation of theoretical relationships with rigor, generality, and simplicity.**Control Theory**

**Control theory**in__control systems engineering__deals with the control of continuously operating__dynamical systems__in engineered processes and machines. The objective is to develop a control model for controlling such systems using a control action in an optimum manner without*delay or overshoot*and ensuring control__stability__.**Artificial Intelligence**

**Artificial intelligence**(**AI**, also**machine intelligence**,**MI**) is intelligence demonstrated by machines, in contrast to the**natural intelligence**(**NI**) displayed by humans and other animals. AI is a blended science ( Science and Mathematics application ). In computer science AI research is defined as the study of "intelligent agents": any device that perceives its environment and takes actions that maximize its chance of successfully achieving its goals. Colloquially, the term "artificial intelligence" is applied when a machine mimics "cognitive" functions that humans associate with other human minds, such as "learning" and "problem solving". The scope of AI is disputed: as machines become increasingly capable, tasks considered as requiring "intelligence" are often removed from the definition, a phenomenon known as the AI effect, leading to the quip, "AI is whatever hasn't been done yet." For instance, optical character recognition is frequently excluded from "artificial intelligence", having become a routine technology. Capabilities generally classified as AI as of 2017 include successfully understanding human speech, competing at the highest level in strategic game systems (such as chess and Go), autonomous cars, intelligent routing in content delivery network and military simulations.**Operations Research**

**Operations research**, or**operational research**in British usage, is a discipline that deals with the application of advanced analytical methods to help make better decisions. Further, the term 'operational analysis' is used in the British (and some British Commonwealth) military as an intrinsic part of capability development, management and assurance. Employing techniques from other mathematical sciences, such as mathematical modeling, statistical analysis, and mathematical optimization, operations research arrives at optimal or near-optimal solutions to complex decision-making problems. Because of its emphasis on human-technology interaction and because of its focus on practical applications, operations research has overlap with other disciplines, notably industrial engineering and operations management, and draws on psychology and organization science. Operations research is often concerned with determining the maximum (of profit, performance, or yield) or minimum (of loss, risk, or cost) of some real-world objective. Originating in military efforts before World War II, its techniques have grown to concern problems in a variety of industries.**Queuing Theory**

**Queueing theory**is the mathematical study of waiting lines, or queues. A queueing model is constructed so that queue lengths and waiting time can be predicted. Queueing theory is generally considered a branch of operations research because the results are often used when making business decisions about the resources needed to provide a service.

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